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Properties

Label 2-55e2-275.156-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.994 - 0.108i$
Analytic cond. $1.50967$
Root an. cond. $1.22868$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.190 − 0.587i)3-s + 4-s + (0.309 + 0.951i)5-s + (0.5 + 0.363i)9-s + (0.190 − 0.587i)12-s + 0.618·15-s + 16-s + (0.309 + 0.951i)20-s + (−0.5 − 1.53i)23-s + (−0.809 + 0.587i)25-s + (0.809 − 0.587i)27-s + (−0.5 + 0.363i)31-s + (0.5 + 0.363i)36-s + (−1.61 + 1.17i)37-s + (−0.190 + 0.587i)45-s + ⋯
L(s)  = 1  + (0.190 − 0.587i)3-s + 4-s + (0.309 + 0.951i)5-s + (0.5 + 0.363i)9-s + (0.190 − 0.587i)12-s + 0.618·15-s + 16-s + (0.309 + 0.951i)20-s + (−0.5 − 1.53i)23-s + (−0.809 + 0.587i)25-s + (0.809 − 0.587i)27-s + (−0.5 + 0.363i)31-s + (0.5 + 0.363i)36-s + (−1.61 + 1.17i)37-s + (−0.190 + 0.587i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1806, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.994 - 0.108i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.874772901\)
\(L(\frac12)\) \(\approx\) \(1.874772901\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 - T^{2} \)
3 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.709716798853847045428046641089, −7.997879021790342264532584100248, −7.15913029109494857735387212824, −6.85368146244868661884518194853, −6.16859242895785841483781332983, −5.27504245327150510569175902076, −4.05431013283386954569831463741, −2.99815413828624000931646052262, −2.30825005750942677392202426799, −1.53742094183483593014882958109, 1.31643053248198658873629243463, 2.16954992976029694707702886035, 3.45373973196267302948170061678, 4.04649758489314656484025367330, 5.16878538612227891302439305669, 5.72045773715099914313151567553, 6.64364540435994549841219011352, 7.42774885731135351512298679097, 8.118125845116305040690614893019, 9.095169854441682226487277719008

Graph of the $Z$-function along the critical line